This way of estimating \(T(P)\)  is taken from empirical risk minimization theory. A particular case of \(T(P)\)  is obtained when choosing \(\psi (x)=x\)  in which case \(T(P)=\mathbb {E}[X]\)  and \(T(X_1^n)=\frac{1}{n}\sum _{i=1}^n X_i\) , however it is well known that the empirical mean is not robust. A careful choice of the function \(\psi \)  yield estimators that are more robust to outliers and to heavy-tailed data (see [1]}).
